Skip to main content
SpringerLink
Log in

Rogers–Ramanujan type identities via Abel’s lemma on summation by parts

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

The Abel’s lemma on summation by parts is employed to review identities of Rogers–Ramanujan type. Twenty examples are illustrated including several new RR identities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, A.K., Bressoud, D.M.: Lattice paths and multiple basic hypergeometric series. Pac. J. Math. 136, 209–228 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1976)

    MATH  Google Scholar 

  3. Andrews, G.E.: Combinatorics and Ramanujans “Lost” Notebook “Sureys in Combinatorics”. Lecture Note Series, vol. 103. London Mathematical Society, pp. 1–23 (1985)

  4. Andrews, G.E.: q-series: Their Development and Application in Analysis Number Theory, Combinatorics, Physics, and Computer Algebra, CBMS Regional Conference Series in Mathematics, vol. 66. American Mathematical Society, Providence, RI, (1986)

  5. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook (Part I). Springer, New York (2005)

    MATH  Google Scholar 

  6. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, vol. Part II. Springer, New York (2005)

    MATH  Google Scholar 

  7. Bailey, W.N.: Some identities in combinatory analysis. Proc. Lond. Math. Soc. (2) 49, 421–435 (1947)

    MathSciNet  MATH  Google Scholar 

  8. Bailey, W.N.: Identities of the Rogers-Ramanujan type. Proc. Lond. Math. Soc. (2) 50, 1–10 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  9. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London and New York (1982)

    MATH  Google Scholar 

  10. Chu, W.: Gould-Hsu-Carlitz inversions and Rogers-Ramanujan identities. Acta Math. Sin. 33(1), 7–12 (1990). MR 91d:05010 & Zbl 727:11037

    MathSciNet  MATH  Google Scholar 

  11. Chu, W.: Abel’s Lemma on summation by parts and Ramanujan’s \(_1\psi _1\)-series identity. Aequationes Math. 72(1/2), 172–176 (2006)

    Article  MathSciNet  Google Scholar 

  12. Chu, W.: Bailey’s very well-poised \({_6\psi _6}\)-series identity. J. Comb. Theory Ser. A 113(6), 966–979 (2006)

    Article  MATH  Google Scholar 

  13. Chu, W.: Abel’s lemma on summation by parts and basic hypergeometric series. Adv. Appl. Math. 39(4), 490–514 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chu, W., Chen, X.J.: Carlitz inversions and identities of Rogers-Ramanujan type. Rocky Mountain J. Math. 44(4), 1125–1143 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chu, W., Wang, C.Y.: The Multisection method for triple products and identities of Rogers-Ramanujan type. J. Math. Anal. Appl. 339(2), 774–784 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chu, W., Wang, C.Y.: Identities of Rogers-Ramanujan type via \(\pm 1\). J. Comput. Anal. Appl. 12(1A), 83–91 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Chu, W., Wang, X.Y.: Telescopic creation for identities of Rogers-Ramanujan type. J. Differ. Equ. Appl. 18(2), 167–183 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chu, W., Zhang, W.L.: Abel’s method on summation by parts and nonterminating \(q\)-series identities. Collect. Math. 60(2), 193–211 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chu, W., Zhang, W.L.: Bilateral Bailey lemma and Rogers-Ramanujan identities. Adv. Appl. Math. 42(3), 358–391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chu, W., Zhang, W.L.: The \(q\)-binomial theorem and Rogers-Ramanujan identities. Utilitas Math. 84, 173–188 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Chu, W., Zhang, W.L.: Four classes of Rogers-Ramanujan identities with quintuple products. Hiroshima Math. J. 41(1), 27–40 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gessel, I., Stanton, D.: Applications of \(q\)-Lagrange inversion to basic hypergeometric series. Trans. Am. Math. Soc. 277(1), 173–201 (1983)

    MathSciNet  MATH  Google Scholar 

  23. Jackson, F.H.: Examples of a generalization of Euler’s transformation for power series. Messenger Math. 57, 169–187 (1928)

    Google Scholar 

  24. Lepowsky, J., Milne, S.C.: Lie algebraic approaches to classical partition identities. Adv. Math. 29(1), 15–59 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mohammed, M.: The \(q\)-Markov-WZ method. Ann. Comb. 9(2), 205–221 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. McLaughlin, J., Sills, A.V.: Ramanujan-Slater type identities related to moduli 18 and 24. J. Math. Anal. Appl. 344, 765–777 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. McLaughlin, J., Sills, A.V., Zimmer, P.: Rogers–Ramanujan–Slater type identities. J. Electron. Comb. 15, #DS15 (2008)

  28. McLaughlin, J., Sills, A.V., Zimmer, P.: Rogers-Ramanujan computer searches. J. Symb. Comput. 44, 1068–1078 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ramanujan, S.: Proof of certain identities in combinatory analysis. Proc. Cambridge Philos. Soc. 19, 214–216 (1919)

    MATH  Google Scholar 

  30. Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25, 318–343 (1894)

    MathSciNet  Google Scholar 

  31. Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. Lond. Math. Soc. 16, 315–336 (1917)

    Article  MATH  Google Scholar 

  32. Sills, A.V.: Finite Rogers–Ramanujan type identities. Electron. J. Comb. 10 (2003), Research Paper#13

  33. Slater, L.J.: Further identities of the Rogers-Ramanujan type. Proc. Lond. Math. Soc. (Ser. 2) 54, 147–167 (1952)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author expresses his sincere gratitude to three anonymous referees for their careful reading, critical comments and valuable suggestions, that have improved the manuscript during the revision.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, 466001, People’s Republic of China

    Wenchang Chu

  2. Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Lecce–Arnesano, P. O. Box 193, 73100, Lecce, Italy

    Wenchang Chu

Authors
  1. Wenchang Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenchang Chu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chu, W. Rogers–Ramanujan type identities via Abel’s lemma on summation by parts. Ramanujan J 48, 459–476 (2019). https://doi.org/10.1007/s11139-018-0087-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-018-0087-7

Keywords

Mathematics Subject Classification

Search

Navigation